Tensor Structures Arising from Affine Lie Algebras

This paper is a part of the series [KL]; however, it can be read independently of the first two parts. In [D3], Drinfeld proved the existence of an equivalence between a tensor category of representations of a quantum group over C[[ ro]] and a tensor category of representations of an undeformed enveloping algebra over C[[ro]] , in which the associativity constraints are given by the Knizhnik-Zamolodchikov equations. However, Drinfeld's proof was nonconstructive, in the sense that he showed that the set of equivalences between the two tensor categories was nonempty, without describing an equivalence explicitly. Our main result is an explicit construction of an equivalence between the tensor categories above. Its main virtue is, as we will show in Part IV, that it "extends analytically" to the case where ro is no longer a formal variable (where one of the two tensor categories becomes the category &'" for an affine Lie algebra as in §2, and the other one becomes a category of representations of a quantum group for a parameter related to K). The relation between Drinfeld's result and ours could be perhaps compared with the relation between Tits's proof of the existence of an isomorphism between the group algebra of a Weyl group and the corresponding Iwahori-Hecke algebra, on the one hand, and the explicit construction [Ll] of such an isomorphism, on the other hand. The tensor categories of interest to us are defined in §§ 19 and 24. In §§20 and 22 we introduce three kinds of basic morphisms Ta , b ' Sc' r i; a, b involving tensor products of "Weyl modules", and we prove various identities relating them. The most complicated such identities are derived from the classical theory of hypergeometric functions. These basic morphisms are combined to define other morphisms in §§21 and 23. They are used in §25 to construct the equivalence of tensor categories.